{"title":"Miraculous cancellations and the quantum Frobenius for $SL_3$ skein modules","authors":"Vijay Higgins","doi":"arxiv-2409.00351","DOIUrl":null,"url":null,"abstract":"We construct a quantum Frobenius map for the $SL_3$ skein module of any\noriented 3-manifold specialized at a root of unity, and describe the map by way\nof threading certain polynomials along links. The homomorphism is a higher rank\nversion of the Chebyshev-Frobenius homomorphism of Bonahon-Wong. The strategy\nbuilds on a previous construction of the Frobenius map for $SL_3$ skein\nalgebras of punctured surfaces, using the Frobenius map of Parshall-Wang for\nthe quantum group $\\mathcal{O}_q(SL_3).$","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"14 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Quantum Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.00351","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We construct a quantum Frobenius map for the $SL_3$ skein module of any
oriented 3-manifold specialized at a root of unity, and describe the map by way
of threading certain polynomials along links. The homomorphism is a higher rank
version of the Chebyshev-Frobenius homomorphism of Bonahon-Wong. The strategy
builds on a previous construction of the Frobenius map for $SL_3$ skein
algebras of punctured surfaces, using the Frobenius map of Parshall-Wang for
the quantum group $\mathcal{O}_q(SL_3).$
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